On the Solution Procedure of Partial Differential Equation (PDE) with the Method of Lines (MOL) Using Crank-Nicholson Method (CNM)

Roknujjaman, Mohammad; Asaduzzaman, Mohammad

doi:10.11648/j.ajam.20180601.11

Cited by 6 publications

(1 citation statement)

References 10 publications

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“…Additionally, this method applied to the equations involves a time variable (t) and one or more space variables (x 1 , x 2 , x m ), transforms them into a system of ordinary differential equations (ODEs) [14]. The part of the equations related to space variables is discredited, resulting in an approximation of the PDE in the form of a system of ODEs.…”

Section: The Methods Of Linesmentioning

confidence: 99%

Numerical Solution of Elliptic Partial Differential Equation: Method of Lines and Crank-Nicholson Method

Roknujjaman,

Arefin,

Joarder

2024

Preprint

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The method of lines (MOL) is a solution procedure for solving partial differential equation (PDE) and the Crank-Nicholson method (CNM) is an implicit finite difference method, used to solve the elliptic equation and similar partial differential equations numerically. In this study, the numerical techniques were introduced to solve elliptic partial differential equation (PDE) together with initial and boundary conditions. The Poisson equation was considered as an elliptic PDE. Particularly, two methods were used to solve a two-dimensional Poisson equation numerically, such as the method of lines (MOL) and Crank-Nicholson method (CNM). The implementation of the solutions was done using Microsoft office spreadsheet and MATLAB programing language. Lastly, the numerical solutions were presented graphically which was obtained with method of lines along with Crank-Nicholson method.

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Section: The Methods Of Linesmentioning

confidence: 99%

Numerical Solution of Elliptic Partial Differential Equation: Method of Lines and Crank-Nicholson Method

Roknujjaman,

Arefin,

Joarder

2024

Preprint

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Experimental and theoretical analysis of pulsatile flow in elastic tubes: Wave propagation and attenuation

Roknujjaman,

Kyotoh,

Yohei

et al. 2023

Physics of Fluids

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The objective of this study was to comprehensively examine the pulsatile flow behavior within an elastic tube using experimental and theoretical approaches. This work is an extension of our earlier report [Roknujjaman et al., Phys. Fluids, 34, 123612 (2022)]. In this paper, we adapted a quasi-one-dimensional model to elucidate the wave propagation and pulse attenuation phenomena in an elastic tube and compared with experimental findings. According to our investigation, the key findings were as follows: the experimental and predicted flow wave propagation trend agreed well. Additionally, because of the silicon tubes greater deformation than the urethane tubes, the pulsation was decreased farther in the silicon tubes than in the urethane ones. Furthermore, because of the reflected wave near the tube exit, the pulsations amplitude significantly increased, but it did not enlarge when the resistance was installed at the exit. Finally, we infer that the quasi-one-dimensional model is a better fit for experimental results compared to the two-dimensional model.

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Analytical Solution of Homogeneous One-Dimensional Heat Equation with Neumann Boundary Conditions

Subani

Jamaluddin

Mohamed

et al. 2020

J. Phys.: Conf. Ser.

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A partial differential equation is an equation which includes derivatives of an unknown function with respect to two or more independent variables. The analytical solution is needed to obtain the exact solution of partial differential equation. To solve these partial differential equations, the appropriate boundary and initial conditions are needed. The general solution is dependent not only on the equation, but also on the boundary conditions. In other words, these partial differential equations will have different general solution when paired with different sets of boundary conditions. In the present study, the homogeneous one-dimensional heat equation will be solved analytically by using separation of variables method. Our main objective is to determine the general and specific solution of heat equation based on analytical solution. To verify our objective, the heat equation will be solved based on the different functions of initial conditions on Neumann boundary conditions. The results have been compared with different values of initial conditions but the boundary condition remain the same. Based on the results obtained, it can be concluded that increase the number of n will reduce the heat temperature and the time taken. For short length of the rod, the heat temperature quickly converges to zero and take less time to release or reduced the heat temperature when compared to the long length of the rod.

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On the Solution Procedure of Partial Differential Equation (PDE) with the Method of Lines (MOL) Using Crank-Nicholson Method (CNM)

Cited by 6 publications

References 10 publications

Numerical Solution of Elliptic Partial Differential Equation: Method of Lines and Crank-Nicholson Method

Numerical Solution of Elliptic Partial Differential Equation: Method of Lines and Crank-Nicholson Method

Experimental and theoretical analysis of pulsatile flow in elastic tubes: Wave propagation and attenuation

Analytical Solution of Homogeneous One-Dimensional Heat Equation with Neumann Boundary Conditions

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